Line Equation: Parallel To 3x-4y=16, Passing Through (4,-4)

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Hey guys! Let's dive into a common problem in coordinate geometry: finding the equation of a line. This isn't just about crunching numbers; it’s about understanding how lines relate to each other on a graph. Specifically, we’re going to figure out the equation of a line that passes through a particular point and is parallel to another line. Sounds like fun, right? Our specific challenge is to find the equation of a line that goes through the point (4, -4) and is parallel to the line 3x - 4y = 16. This type of problem pops up all the time in algebra and geometry, so mastering it is super useful.

Understanding Parallel Lines and Their Slopes

So, first things first: what does it mean for lines to be parallel? In simple terms, parallel lines are like train tracks; they run in the same direction and never intersect. The key concept here is that parallel lines have the same slope. Slope, remember, is the measure of how steep a line is. It tells us how much the line rises (or falls) for every unit we move horizontally. This is often referred to as "rise over run."

To kick things off, we need to figure out the slope of the line we're given: 3x - 4y = 16. The easiest way to do this is to rearrange the equation into what's called slope-intercept form. This form looks like this: y = mx + b. In this equation, 'm' is the slope, and 'b' is the y-intercept (where the line crosses the y-axis). Trust me, getting comfortable with slope-intercept form makes life so much easier when you're dealing with lines!

Let's transform 3x - 4y = 16 into slope-intercept form. Here's how we do it step-by-step:

  1. Start with the original equation: 3x - 4y = 16.
  2. Subtract 3x from both sides: -4y = -3x + 16.
  3. Divide both sides by -4: y = (3/4)x - 4.

Now, we can clearly see that the slope ('m') of the given line is 3/4. Since our target line is parallel to this one, it also has a slope of 3/4. This is a critical piece of information because now we know the 'm' value for our new line. Remember, parallel lines share the same slope, which is the cornerstone of solving this type of problem. Identifying and understanding this shared slope is often the first major step in finding the equation of a parallel line. It sets the stage for the rest of the solution, allowing us to focus on finding the specific equation that also satisfies the given point.

Using Point-Slope Form to Find the Equation

Alright, so we've nailed down the slope of our mystery line – it's 3/4, just like the line it's parallel to. But how do we actually find the full equation of the line? This is where the point-slope form comes to the rescue! The point-slope form is a super handy way to write the equation of a line when you know a point on the line (x₁, y₁) and the slope (m). It looks like this: y - y₁ = m(x - x₁).

Why is this so useful? Well, we already have both the slope (m = 3/4) and a point that our line passes through: (4, -4). That's all the info we need to plug into the point-slope form! Let's do it:

  1. Write down the point-slope form: y - y₁ = m(x - x₁).
  2. Substitute the slope (m = 3/4) and the point (4, -4): y - (-4) = (3/4)(x - 4).

Notice how we carefully plugged in the values, paying close attention to the signs. A negative sign can easily trip you up, so double-check everything! Now, let's simplify this equation:

  • y + 4 = (3/4)(x - 4)

This is a perfectly valid equation for our line, but often, we want to get it into the familiar slope-intercept form (y = mx + b) or standard form (Ax + By = C). So, let's take it a step further and convert it.

Converting to Slope-Intercept Form

Now that we have the equation in point-slope form: y + 4 = (3/4)(x - 4), let's make it look even cleaner by converting it to slope-intercept form (y = mx + b). This form is super useful because it immediately tells us the slope (m) and the y-intercept (b) of the line. Plus, it's often the way equations are presented in textbooks and on tests, so it's a good skill to have.

Here's how we'll do it, step-by-step:

  1. Start with our point-slope equation: y + 4 = (3/4)(x - 4).
  2. Distribute the (3/4) on the right side: y + 4 = (3/4)x - 3.
  3. Subtract 4 from both sides to isolate 'y': y = (3/4)x - 7.

Boom! We've done it. Our equation is now in slope-intercept form: y = (3/4)x - 7. We can clearly see that the slope (m) is 3/4 (which we already knew, since it's parallel to the given line), and the y-intercept (b) is -7. This means the line crosses the y-axis at the point (0, -7).

Slope-intercept form gives us a quick snapshot of the line's key characteristics. It's like having a secret decoder ring for lines! But sometimes, you might need the equation in a different format, like standard form. So, let's explore how to get there too.

Converting to Standard Form

Okay, so we've got our equation in slope-intercept form: y = (3/4)x - 7. That's awesome, but sometimes you'll need to express the equation in standard form, which looks like this: Ax + By = C. Standard form has its own advantages, especially when you're dealing with systems of equations or want a clean, integer-based representation.

Here’s the lowdown on converting to standard form:

The main goal is to get rid of fractions and rearrange the equation so that 'x' and 'y' terms are on one side and the constant is on the other, with 'A' being a positive integer. Here’s how we tackle it:

  1. Start with our slope-intercept form: y = (3/4)x - 7.
  2. Get rid of the fraction by multiplying every term on both sides by 4: 4y = 3x - 28.
  3. Move the 'x' term to the left side by subtracting 3x from both sides: -3x + 4y = -28.
  4. Now, standard form usually prefers the coefficient of 'x' (which is 'A') to be positive. So, we'll multiply the entire equation by -1: 3x - 4y = 28.

There we have it! The equation of our line in standard form is 3x - 4y = 28. Notice how it neatly fits the Ax + By = C format. We've successfully banished the fraction and made the 'x' coefficient positive. Converting to standard form might seem like an extra step, but it's a useful skill for more advanced math topics.

Putting It All Together

Wow, we've covered a lot! Let's take a moment to recap the entire process of finding the equation of a line that passes through the point (4, -4) and is parallel to the line 3x - 4y = 16.

  1. Find the slope of the given line: We rearranged 3x - 4y = 16 into slope-intercept form (y = (3/4)x - 4) and identified the slope as 3/4.
  2. Use the same slope for the parallel line: Since parallel lines have the same slope, our target line also has a slope of 3/4.
  3. Use point-slope form: We plugged the slope (m = 3/4) and the point (4, -4) into the point-slope form: y - (-4) = (3/4)(x - 4).
  4. Convert to slope-intercept form (optional): We simplified the equation to y = (3/4)x - 7, which clearly shows the slope and y-intercept.
  5. Convert to standard form (optional): We further transformed the equation to 3x - 4y = 28, a clean, integer-based representation.

So, the equation of the line that passes through (4, -4) and is parallel to 3x - 4y = 16 can be expressed in a few ways:

  • Point-slope form: y + 4 = (3/4)(x - 4)
  • Slope-intercept form: y = (3/4)x - 7
  • Standard form: 3x - 4y = 28

Each form is equally valid, and the best one to use often depends on the specific situation or what you're asked for in a problem.

Understanding the connections between these different forms is key to mastering linear equations. It’s not just about getting the right answer; it's about understanding the underlying concepts. The ability to move fluidly between point-slope, slope-intercept, and standard forms is a hallmark of a confident math student. It allows you to tackle a wider range of problems and demonstrate a deeper understanding of the material.

So, next time you encounter a problem like this, remember the steps we’ve covered. Think about parallel lines and their slopes, embrace the point-slope form, and don't be afraid to convert to other forms as needed. You've got this!

In conclusion, by understanding the properties of parallel lines and mastering the different forms of linear equations, you can confidently tackle these types of problems. The equation 3x - 4y = 28 (in standard form) represents the line that meets the given criteria. Keep practicing, and you'll become a pro at linear equations in no time!